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Sharpening the Becker-Stark Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 931275 (2010)
In this paper, we establish a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one.
Theorem 1.1 (see [1, Lemma ]).
If , then
Let , then
Furthermore, and are the best constants in (1.2).
In fact, we can obtain the following further results.
Let , then
Furthermore, and are the best constants in (1.3).
In this paper, in the form of (1.2) and (1.3) we shall show a general refinement of the Becker-Stark inequalities as follows.
Let , and let be a natural number. Then
holds, where , and
where are the even-indexed Bernoulli numbers.
Furthermore, and are the best constants in (1.4).
2. Four Lemmas
The function is decreasing, where is Riemann's zeta function.
is equivalent to the function , which is decreasing.
Lemma 2.2 (see [5, Theorem ]).
Let be Riemann's zeta function and the even-indexed Bernoulli numbers. Then
Lemma 2.3 (see [6, 220.127.116.11 (1.3)]).
Let . Then
Let and . Then , where
By Lemma 2.3, we have
Since is decreasing by Lemma 2.1, it follows that
From Lemma 2.2, we get
which implies that for .
3. Proofs of Theorems
Proof of Theorem 1.4.
By Lemma 2.4, we have for , and is decreasing on .
At the same time, = by (3.1), and by (3.2), so and are the best constants in (1.4).
Proof of Theorem 1.3.
Let in Theorem 1.4; we obtain that and . Then the proof of Theorem 1.3 is complete.
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Zhu, L., Hua, J. Sharpening the Becker-Stark Inequalities. J Inequal Appl 2010, 931275 (2010). https://doi.org/10.1155/2010/931275
- Power Series
- Natural Number
- Series Expansion
- Zeta Function
- Power Series Expansion